System and method of semiconductor characterization

ABSTRACT

A method for characterizing a semiconductor sample, said method comprising: shining light on one or more points in said semiconductor sample; measuring one or more voltage decay curves corresponding to said shining of light on said one or more points in said semiconductor sample; extracting one or more intermediate voltage decay curves corresponding to one or more measured voltage decay curves; obtaining one or more normalized decay curves corresponding to one or more intermediate voltage decay curves, each of the said one or more normalized decay curves corresponding to one or more discrete estimates of survival functions; and analyzing said obtained one or more normalized decay curves, said analyzing comprising obtaining one or more discrete estimates of the probability of recombination corresponding to the one or more normalized decay curves, and computing one or more summary statistics corresponding to each of said obtained one or more discrete estimates.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of and claims priority toU.S. patent application Ser. No. 14/336,046 filed on Jul. 21, 2014; andclaims the benefit of U.S. Provisional Patent Application No.62/019,460, filed on Jul. 1, 2014, each of which is incorporated hereinby reference in its entirety.

FIELD OF THE INVENTION

The present disclosure relates to characterization of semiconductors.

BRIEF SUMMARY

A method for characterizing a semiconductor sample, said methodimplemented using an analysis system comprising: a transientphotoconductive decay measurement subsystem, a database, a data analysissubsystem, and a statistical analysis subsystem, said transientphotoconductive decay measurement subsystem, database, data analysissubsystem and statistical analysis subsystem connected to each other byan interconnection, said method comprising the steps of: shining, usingthe transient photoconductive decay measurement subsystem, light on oneor more points in said semiconductor sample; measuring, using thetransient photoconductive decay measurement subsystem, one or morevoltage decay curves corresponding to said shining of light on said oneor more points in said semiconductor sample; extracting, using the dataanalysis subsystem, one or more intermediate voltage decay curvescorresponding to one or more measured voltage decay curves; obtaining,using the data analysis subsystem, one or more normalized decay curvescorresponding to one or more intermediate voltage decay curves, each ofthe said one or more normalized decay curves corresponding to one ormore discrete estimates of survival functions; analyzing, using thestatistical analysis subsystem, said obtained one or more normalizeddecay curves, said analyzing comprising obtaining one or more discreteestimates of the probability of recombination corresponding to the oneor more normalized decay curves, and computing one or more summarystatistics corresponding to each of said obtained one or more discreteestimates; and determining, by either the statistical analysis subsystemor the data analysis subsystem, the presence of nonuniformities withinsaid semiconductor sample based on results of said analyzing.

A method for characterizing a semiconductor sample, said methodcomprising: measuring a plurality of voltage decay curves correspondingto a plurality of points in said semiconductor sample; extracting aplurality of intermediate voltage decay curves corresponding to saidplurality of measured voltage decay curves; converting said extractedplurality of intermediate voltage decay curves to a plurality ofminority carrier population decay curves; performing one or morecomparisons of survival behavior using said plurality of minoritycarrier population decay curves; and determining presence ofnonuniformities within said semiconductor sample using said results fromsaid performing of one or more comparisons of survival behavior.

A method for comparing a plurality of semiconductor samples, said methodcomprising: measuring a plurality of voltage decay curves correspondingto a plurality of points from said plurality of semiconductor samples;extracting a plurality of intermediate voltage decay curvescorresponding to said plurality of measured voltage decay curves;converting said extracted plurality of intermediate voltage decay curvesto a plurality of minority carrier population decay curves; performingone or more comparisons of survival behavior using said plurality ofminority carrier population decay curves; and determining presence ofnonuniformities within said plurality of semiconductor samples usingsaid results from said performing of one or more comparisons of survivalbehavior.

The foregoing and additional aspects and embodiments of the presentdisclosure will be apparent to those of ordinary skill in the art inview of the detailed description of various embodiments and/or aspects,which is made with reference to the drawings, a brief description ofwhich is provided next.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other advantages of the disclosure will becomeapparent upon reading the following detailed description and uponreference to the drawings.

FIG. 1 shows the steps used in the prior art analysis approaches

FIG. 2 shows an example analysis setup.

FIG. 3 shows a sample voltage decay curve 300.

FIG. 4 shows a grid of points for mapping spatial nonuniformities acrossa semiconductor sample.

FIG. 5A shows one embodiment of an analysis performed on a voltage decaycurve.

FIG. 5B shows an example of extracting readings to form an intermediatevoltage decay curve V(t).

FIG. 6 shows one embodiment of step 504.

FIG. 7 shows an embodiment of a process to perform comparison betweentwo or more samples, or two or more locations within the same sample.

FIG. 8 shows an example flowchart to calculate and compare summarystatistics.

FIG. 9 shows a process to estimate an upper bound t_(win) to calculatethe proportion of recombination events taking place within a localizedregion of interest compared to the number of recombination events takingplace within the entire semiconductor sample.

FIG. 10 shows a process to estimate an upper bound t_(win) to calculatethe proportion of minority carriers within the localized area ascompared to the entire semiconductor sample.

While the present disclosure is susceptible to various modifications andalternative forms, specific embodiments or implementations have beenshown by way of example in the drawings and will be described in detailherein. It should be understood, however, that the disclosure is notintended to be limited to the particular forms disclosed. Rather, thedisclosure is to cover all modifications, equivalents, and alternativesfalling within the spirit and scope of an invention as defined by theappended claims.

DETAILED DESCRIPTION

While the description here focuses on analysis of results obtained usingthe transient photoconductive decay technique, it is equally applicableto the analysis of results obtained using other similar techniques, thatis, where a population is generated, and a measurable output related tothe survival of the generated population is measured.

Similarly while many of the examples here are discussed in relation tomercury cadmium telluride (HgCdTe), the analyses are equally applicableto any semiconductor material.

Introduction

In the transient photoconductive decay technique, in one embodiment acurrent is passed through a sample which is to be measured. As a result,a voltage is created across the sample. The voltage is proportional tothe resistance of the sample, which in turn is proportional to theresistivity and hence inversely proportional to the conductivity of thesample.

Then, light is shone upon the sample to generate electron hole pairs.Depending on whether the sample is p-type or n-type, either electrons orholes are the minority carriers. As a result, the conductivity of thesample increases, leading to a drop in the resistance of the sample. Asa consequence of the resistance drop, the voltage across the sample willalso drop.

The generated minority carriers drift under the influence of the biasfield created by the voltage and diffuse throughout the sample due tothe concentration gradient. Over time, these minority carriers will alsorecombine within the sample. As a consequence of the recombination ofthe minority carriers, the conductivity will decay to its value beforethe light was shone upon the sample. As the conductivity decays so doesthe resistance, and consequently the voltage across the sample will alsoincrease.

The aim of the transient photoconductive decay technique is to analysethe decay of the induced transient increase in conductivity by measuringthe change in voltage across the sample. The sample can be characterizedusing this technique.

A variation of the transient photoconductive decay technique is spatialmapping. In this variation, light is shone on different locations of thesample, and the resultant transient photoconductive decay curves aremeasured. By doing so, spatial variations across a sample can becharacterized.

Material Parameters

In previous works, several parameters of interest have beencharacterized using the transient photoconductive decay technique. Someof these parameters are described in section 1.3 of R. Rajaduray,“Investigation of Spatial Characterisation Techniques inSemiconductors,” Honours Thesis 1998, University of Western Australia,herein incorporated by reference as if reproduced in its entirety.

Two parameters of interest which have been characterized in previousworks are the bulk minority carrier lifetime and the surfacerecombination velocity.

The bulk minority carrier lifetime τ_(b) is the average time a minoritycarrier identified at a particular instant and location within the bulkof a semiconductor will exist until recombination. It is defined by:

$\tau_{b} = \frac{p - p_{o}}{R}$

where

-   -   p is the total minority carrier density    -   p_(o) is the equilibrium minority carrier density    -   R is the minority carrier recombination rate

The bulk minority carrier lifetime τ_(b) is highly dependent upon thenature of the recombination mechanisms within the bulk of thesemiconductor.

The surface recombination velocity s is a measure of the recombinationrate of minority carriers at the surface of a semiconductor. It isdefined for excess holes in an n-type semiconductor with a surface atx=0 by:

$s = \left. {D_{a}\frac{\partial p}{\partial x}\frac{1}{p}} \right|_{x = 0}$

where

-   -   D_(a) is the ambipolar diffusion coefficient    -   p is the excess hole concentration

For excess electronics in a p-type semiconductor with a surface at x=0:

$s = \left. {D_{a}\frac{\partial n}{\partial x}\frac{1}{n}} \right|_{x = 0}$

where

-   -   D_(a) is the ambipolar diffusion coefficient    -   n is the excess hole concentration

Physically, the surface recombination velocity can be understood asfollows: A current of holes or electrons of density p or n drift with anaverage velocity equal to the surface recombination velocity s into thesurface and the holes or electrons are then removed. Thus, as thesurface recombination velocity increases, the excess hole or electronconcentration at the surface decreases.

Recombination Mechanisms

A detailed explanation of examples of various bulk recombinationmechanisms in, for example, HgCdTe is given in Sections 2.2.1 to 2.2.3of R. Rajaduray, “Investigation of Spatial Characterisation Techniquesin Semiconductors,” Honours Thesis 1998, University of WesternAustralia.

For example, with reference to HgCdTe three important bulk recombinationmechanisms are Auger, radiative and Shockley-Read-Hall (SRH)recombination. Auger and radiative recombination are strongly dependentupon the carrier concentrations and energy gap. SRH recombination isassociated with the presence of defect states within the bandgap, knownas traps.

There may be other bulk recombination mechanisms present in othersemiconductor materials.

Similarly, an explanation of surface recombination mechanisms in HgCdTeis given in section 2.3 of R. Rajaduray, “Investigation of SpatialCharacterisation Techniques in Semiconductors,” Honours Thesis 1998,University of Western Australia.

Three surface recombination mechanisms in HgCdTe are:

-   -   Thermal transitions through Shockley-Read-Hall centres in the        depletion region: This process is similar to the SRH bulk        recombination mechanism.    -   Thermal transitions via fast surface states.    -   Tunnel transitions through the Shockley-Read-Hall centres in the        depletion region

There may be other surface recombination mechanisms present in othersemiconductor materials.

Previous Analysis Approaches

Many of the existing analysis approaches are based on parametrictechniques. FIG. 1 shows the steps involved in the prior art analysisapproaches:

-   -   101: Creating a model based on one or more assumptions    -   102: Setting up one or more differential equations with boundary        conditions based on the assumptions    -   103: Solving the differential equations to obtain a solution        with one or more parameters which shows the expected behavior of        the decay of the minority carrier population over time, and    -   104: Fitting experimental results to the obtained solution        using, for example, least squares fitting to extract the one or        more solution parameters.

Two examples of steps 101-103 are explained below. The first exampleuses the approach detailed in W. Van Roosbroeck, “Injected CurrentCarrier Transport in a Semi-Infinite Semiconductor and the Determinationof Lifetimes and Surface Recombination Velocities.” Journal of AppliedPhysics 26.4 (1955): 380-391. The solution is given as:

p(U)=p(0)exp[U(S ²−1)]erfc[S√{square root over (U)}]  (1)

where

-   -   U is time t normalized with respect to τ_(b)    -   τ_(b) is the bulk minority carrier lifetime    -   p(U) is the minority carrier population at normalized time U or        at time t=U×τ_(b)    -   p(0) is the minority carrier at normalized time U=0 or        equivalently t=0    -   S is the normalized surface recombination velocity. S is further        given by:

$S = \frac{s\; \tau_{b}}{L}$

where

-   -   s is the surface recombination velocity    -   L is the minority carrier diffusion length, given by        √(D_(a)×τ_(b))

The second example of steps 101-103 provides a solution for a finiterectangular sample of dimensions 2A, 2B and 2C and uses the approachdetailed in J. S. Blakemore, Semiconductor Statistics, Oxford Pergamon1962. The solution is given as a series of eigenfunctions for differentmodes (i,j,k) and is given by:

$\begin{matrix}{{p(t)} = {\frac{p(0)}{ABC}{\sum\limits_{ijk}{K_{ijk} \times {\exp \left\lbrack {- {t\left( {v_{b} + v_{ijk}} \right)}} \right\rbrack}}}}} & (2)\end{matrix}$

where

-   -   p(t) is the minority carrier population at time t    -   p(0) is the minority carrier at time t=0    -   K_(ijk) is the constant for mode (i, j, k)    -   ν_(ikj) is the inverse of the time constant for mode (i, j, k)    -   ν_(b) is the inverse of the bulk minority carrier lifetime τ_(b)

Then, once a solution such as in equations (1) and (2) above have beenprovided, experimentally obtained decay curves can be fitted to thesecurves using, for example, least squares regression. The parameters usedto obtain the best fit are extracted and recorded. For example, usingthe Van Roosbroeck model, the surface recombination velocity s and bulkminority carrier lifetime τ_(b) to obtain the best fit are extracted.

There are other analysis approaches which are variations of these 2approaches. Usually, these variations employ slightly differentassumptions to create a model. However many of these approaches areflawed for several reasons.

Many of the existing approaches use models which employ unrealisticassumptions and then set up differential equations and boundaryconditions based on these unrealistic conditions.

For example, firstly many of the models assume that recombinationparameters such as the bulk minority carrier lifetime and the surfacerecombination velocity are spatially and temporally constant within theanalyzed semiconductor sample. This has clearly been shown not to be thecase. Studies such as those performed by V. C. Lopes et at“Characterization of (Hg, Cd)Te by the Photoconductive Decay Technique,”J. Vac Sci, vol. 8, no. 2, pp. 1167-1170, March/April 1990; and R. G.Pratt et at “Minority carrier lifetime in n-type Bridgman grownHg_(1-x)Cd_(x)Te,” J. Appl. Physics vol 54, no. 9 pp. 5152-5157, 1983;showed spatial nonuniformity of bulk lifetime across semiconductorsamples. Furthermore, as shown in Chapter 6 of Ramesh Rajaduray,“Investigation of Spatial Characterisation Techniques inSemiconductors,” Honours Thesis 1998, University of Western Australia,the extracted parameters clearly exhibited temporal nonuniformity, thatis, when segments of a voltage decay curve with differing temporalextents were fitted to equation (1), the values of the extractedparameters were non-uniform.

Secondly, many of the models assume that the dominant recombinationmechanism is independent of the minority carrier density. This is alsounrealistic, when it has been shown in that in certain situations,minority carrier concentration dependent recombination mechanisms suchas Auger recombination will dominate in materials such as HgCdTe. As anexample, in pages 53 and 54 of Ramesh Rajaduray, “Investigation ofSpatial Characterisation Techniques in Semiconductors,” Honours Thesis1998, University of Western Australia, it was shown that at time t=0, apopulation of minority carriers equivalent to 19% of the total number ofminority carriers is generated within a small area. It was shown in, forexample, G. Nimtz, et al. “Transient carrier decay and transportproperties in Hg_(1-x)Cd_(x)Te.” Phys. Rev. BIO p 3302 (1974); and F.Bartoli et al. “Auger-limited carrier lifetimes in HgCdTe at high excesscarrier concentrations.” Journal of Applied Physics vol. 45 no. 5 pp.2150-2154 (1974); that under such conditions Auger recombination islikely to dominate over radiative and Shockley-Read-Hall mechanisms.

Furthermore, as was pointed out by D. A. Redfern et al “On the transientphotoconductive decay technique for lifetime extraction in HgCdTe” inOptoelectronic and Microelectronic Materials Devices, 1998. Proceedings.1998 Conference on, pp. 275-278. IEEE, 1999, “none of the current modelsunambiguously [explained] experimental results and that detailedlifetime extraction by photoconductive decay is still not a quantitativetechnique.”

In addition, the generation and recombination of minority carriers whichoccur within a semiconductor sample each time light is incident on thesample, are random processes. As a consequence, carrier concentrationsat particular points within a semiconductor sample are also likely tovary randomly as well. This means that diffusion based movements, whichare highly dependent on concentration gradients, are also likely to berandom in nature. As a consequence, this further intensifies the randombehavior of the carrier concentration at a particular point within asemiconductor sample. If carrier concentration dependent recombinationmechanisms dominate, then the random behavior is even furtherintensified. However many of the differential equations set up in steps101-104 of FIG. 1 above are assumed to be deterministic in nature.

As a consequence of the above, many of the previously proposed modelsemploy unrealistic assumptions which lead to an incorrect understandingof the evolution of the population of generated minority carriers overtime within a semiconductor sample.

As a further consequence, analysis approaches which use such models toperform spatial mapping of recombination parameters, such as, forexample, spatial bulk minority carrier lifetime mapping are inherentlyflawed. Not only is the understanding of the evolution of the populationover time wrong, but the incorrect behavior is then used to detectparameter variations which is fundamentally opposite to the assumptionsemployed.

Therefore there is a need for analysis approaches which are less relianton using models with inherently unrealistic assumptions to performparametric-based analysis, or worse still: Using models built on certainassumptions with the aim of detecting properties which are in directopposition to these assumptions.

New Analysis Approaches

This section demonstrates several analysis approaches which overcome theproblems due to the parametric analysis approaches used previously.

An example analysis setup is shown in FIG. 2. Transient photoconductivedecay measurement subsystem 201 is used to obtain voltage decay curvesfor a semiconductor sample. In one embodiment, transient photoconductivedecay measurement subsystem 201 is similar to that detailed in section5.3 of Ramesh Rajaduray, “Investigation of Spatial CharacterisationTechniques in Semiconductors,” Honours Thesis 1998, University ofWestern Australia. In another embodiment, transient photoconductivedecay measurement subsystem 101 is similar to that used in Pratt et at“Minority carrier lifetime in n-type Bridgman grown Hg_(1-x)Cd_(x)Te”Journal of applied physics 54, no. 9 (1983): 5152-5157, and hereinincorporated by reference in its entirety. A further example is given inT. Tomlin, “Spatial Mapping of Minority carrier lifetime in MercuryCadmium Telluride,” Honours Thesis 1995, University of WesternAustralia. In one embodiment, as shown in FIG. 3, an obtained voltagedecay curve 300 denoted as V_(me)(t) comprises a plurality ofmeasurements comprising measurements 305, 306 and 307 of the voltageacross the sample at corresponding times 301, 303, and 304. The timeinstant corresponding to each measurement within the plurality isseparated from the time instant corresponding to the precedingmeasurement by a time interval Δt (302), such as shown in FIG. 3.

Statistical analysis subsystem 204 performs statistical analyses whichwill be described later. In one embodiment, the statistical analysissubsystem 204 is implemented in hardware. In one embodiment, thestatistical analysis subsystem 204 is implemented in software. In yetanother embodiment, statistical analysis subsystem 204 is implemented ina combination of hardware and software. Different programming languagesand systems can be used to implement statistical analysis subsystem 104,including, for example, SPSS, S, R, STATA, Matlab™, SAS, Microsoft™Excel™, SQL and C++.

Data analysis subsystem 203 performs various functions, includingpreparing data for statistical analysis subsystem 204, collating theresults of analysis performed by statistical analysis subsystem 204,performing further analysis of the results from statistical analysissubsystem 204 and presenting the results of these analyses. In oneembodiment, the data analysis subsystem 203 is implemented in hardware.In one embodiment, the data analysis subsystem 203 is implemented insoftware. In yet another embodiment, data analysis subsystem 203 isimplemented in a combination of hardware and software. Differentprogramming languages and systems can be used to implement statisticalanalysis subsystem 204, including, for example, SPSS, S, R, STATA,Matlab™, SAS, Microsoft™ Excel™, SQL and C++.

Database 205 is used to store voltage decay curve data obtained fromtransient photoconductive decay subsystem 201, and data for intermediateprocessing performed by statistical analysis subsystem 204, dataanalysis subsystem 203. Different programming languages and systems canbe used to implement statistical analysis subsystem 204, including, forexample, SQL and Microsoft™ Access™.

Interconnection 202 is used to connect the different subsystemstogether. These could include, for example, local area networks (LAN),campus area network (CAN), wide area networks (WAN). Interconnection 202could encompass one or more subnetworks. Interconnection 202 could beimplemented using various media including wireless, wired, opticalnetwork, and could encompass various technologies including Ethernet andIP-based networks.

In one embodiment, the system illustrated in FIG. 2 is used to compareone or more semiconductor samples. Then, for each of the one or moresemiconductor samples, one or more voltage decay curves such as voltagedecay curve 300 in FIG. 3, is measured using, for example, transientphotoconductive decay measurement subsystem 201.

In another embodiment, the system illustrated in FIG. 2 is used to mapspatial non-uniformities across a single semiconductor sample. In thisembodiment, using transient photoconductive decay measurement subsystem201, light is shone at different points across a semiconductor sample,and for each point a voltage decay curve is obtained. For example, inone embodiment, light is shone at each point, for example points 401,402 and 403 within a grid of points 404 such as shown in FIG. 4 forsample 400 is created. Voltage decay curves such as voltage decay curve300 of FIG. 3 as shown above are then obtained for each point.

As explained previously, the generation, movement and recombination ofminority carriers which occurs within a semiconductor sample each timelight is incident on a spot on the semiconductor sample using thetransient photoconductive decay measurement subsystem 201 are randomsub-processes which are part of a single overall random or stochasticprocess. Consequently, each obtained voltage decay curve represents theevolution of the population of minority carriers with time for onerealization of this overall random process.

In one embodiment, in order to remove the impact of noise, light isshone on the same spot on the semiconductor sample a plurality of times.Then, each time light is shone on the spot, a corresponding voltagedecay curve is obtained.

In one embodiment, the following analysis is applied to each obtainedvoltage decay curve as shown in FIG. 5A.

In step 501, using for example, data analysis subsystem 203, the segmentof the measured voltage decay curve with times greater than the timecorresponding to the peak of the voltage decay curve is extracted toform an intermediate voltage decay curve V(t). An example is shown inFIG. 5B. The time 5A-01 corresponds to the peak (5A-02) of the obtainedvoltage decay curve 5A-00. Then, the segment 5A-03 of the voltage decaycurve 5A-00 for all times greater than time 5A-01 is extracted, to formintermediate voltage decay curve 5A-04. Each time instant onintermediate voltage decay curve 5A-04 is separated from the next timemeasurement by Δt (5A-08), such as, for example, time instants 5A-05,5A-06 and 5A-07. The intermediate voltage decay curve 5A-04 can berepresented as V(nΔt), n=0, 1, 2 . . . N where nΔt are the timeinstants.

In the embodiment where light is shone on the same spot a plurality oftimes and a corresponding measured voltage decay curve is obtained foreach time, a plurality of corresponding intermediate voltage decaycurves V_(k)(nΔt) are obtained, k=1, 2, 3 . . . K. The correspondingintermediate voltage decay curves are then averaged out to provide asmoothed intermediate voltage decay curve V_(s)(nΔt), that is:

${V_{S}\left( {n\; \Delta \; t} \right)} = \frac{\Sigma_{k = 1}^{K}{V_{k}\left( {n\; \Delta \; t} \right)}}{K}$

This is performed for n=0, 1, 2 . . . N.

In optional step 502, using for example, data analysis subsystem 203,the intermediate voltage decay curve V(nΔt) or smoothed intermediatevoltage decay curve V_(s)(nΔt) obtained in step 501 is converted to aminority carrier population decay curve p(t). Various approaches toperform this conversion are known to those of skill in the art and willnot be explained further within this specification.

In step 503, in one embodiment, using for example, data analysissubsystem 203 the intermediate voltage decay curve V(nΔt) or smoothedintermediate voltage decay curve V_(s)(nΔt) obtained in step 501 isnormalized to the voltage at time t=0 to obtain a normalized decay curveV_(no)(nΔt). In an alternate embodiment, if optional step 502 isperformed, the p(t) obtained in step 502 is normalized to the minoritycarrier population at time t=0 to obtain a normalized decay curvep_(no)(t).

The normalized decay curve represents a discrete estimate S_(e)(nΔt),n=0, 1, 2 . . . N of the continuous time survival function S(t)=P[τ>t].S(t) is the probability that minority carriers will survive, that is notrecombine, until beyond time t. If, in step 501, a smoothed intermediatevoltage decay curve is provided as an output, then the obtainednormalized smoothed decay curve is a better discrete estimate S_(e)(t)of S(t).

In step 504, S_(e)(t) is analysed using, for example, statisticalanalysis subsystem 204. One embodiment of step 504 is shown in FIG. 6.The cumulative distribution function CDF(t)=P[τ≦t] is obtained byP[τ≦t]=1−S(t). In an optional embodiment, in step 601, a discreteestimate CDF_(e)(nΔt) of the cumulative distribution function CDF(t) isobtained by computing 1−S_(e)(nΔt), n=0, 1, 2 . . . .

In step 602, in one embodiment, an estimate of the probability ofrecombination between time [nΔt] and [(n+1)Δt], n=0, 1, 2 . . . isobtained. The probability is given by the probability mass functionPMF_(e)[(n+1)Δt], which is obtained by taking successive differences ofthe estimate of the survival function S_(e)(t). That is, the estimatePMF_(e)[(n+1)Δt], n=0, 1, 2 . . . is given by S_(e)[nΔt]−S_(e)[(n+1)Δt].Alternatively, if in step 603, CDF_(e)(nΔt) is calculated, thenPMF_(e)[(n+1)Δt] is given by CDF_(e)[(n+1)Δt]−CDF_(e)[nΔt].

In step 603, one or more summary statistics are computed. In oneembodiment, an estimate of the mean of τ denoted as E[τ] or μ_(τ) iscalculated. In one embodiment, S_(e)(t) is used directly to calculateE[τ] performing the summation of S_(e)(t) from n=1 onwards. In anotherembodiment PMF_(e)[(n+1)Δt] is used.

In another embodiment, the variance of τ denoted as Var(τ) oralternatively σ_(τ) ² is computed.

Other summary statistical computations can be performed including momentgeneration, Laplace transform and characteristic function generation.Moments can also be calculated. Other expectations can also becalculated using the generalized formulae such as, for example E(1/τ³)and E(1/τ²).

In another embodiment, in step 604, one or more survival statisticalcomputations are applied. In one embodiment, the discrete time hazardprobability λ_(e)[(n+1)Δt], which is the probability of recombinationfor a minority carrier between times [nΔt] and [(n+1)Δt] given that theminority carrier has not recombined before time [nΔt] is calculated.Mathematically this is given by:

${\lambda_{e}\left\lbrack {\left( {n + 1} \right)\Delta \; t} \right\rbrack} = \frac{{PMF}_{e}\left\lbrack {\left( {n + 1} \right)\Delta \; t} \right\rbrack}{S_{e}\left( {n\; \Delta \; t} \right)}$

Alternatively it can be calculated as:

${\lambda_{e}\left\lbrack {\left( {n + 1} \right)\Delta \; t} \right\rbrack} = {1 - \frac{S_{e}\left\lbrack {\left( {n + 1} \right)\Delta \; t} \right\rbrack}{S_{e}\left( {n\; \Delta \; t} \right)}}$

Quantiles of S_(e)(nΔt) can be calculated as well. For example, thelowest decile of S_(e)(nΔt), that is, the time after which 90% of theminority carrier population at t=0 have not recombined can be calculatedby determining when S_(e)(nΔt) drops below 0.90. Similarly the highestquartile of S_(e)(nΔt), that is, the time after which 25% of theminority carrier population at t=0 have not recombined can be calculatedby determining when S_(e)(nΔt) drops below 0.25.

Another survival statistical computation which can be performed iscalculating the mean residual time E(τ−nΔt|τ≧nΔt). This gives theexpected time until recombination for a minority carrier, given that theminority carrier survived up to time nΔt. This can be calculated usingwell known mathematical formulas and will not be discussed in detailwithin this specification.

It may be necessary to compare minority carrier decay behavior for twoor more samples, or at two or more locations within the same sample, todetermine if there is nonuniformity between samples or whether there isspatial nonuniformity within the same sample. Then, known mathematicaltechniques to compare survival behaviour for different populations ofgenerated minority carriers can be employed. This involves using aprocess similar to that outlined in FIG. 5A, except that step 503 is notperformed. An embodiment is shown in FIG. 7. Steps 701 and 702 areidentical to steps 501 and 502, except that step 502 is not optional.These two steps are performed for every sample, or for every point orlocation within the same sample. These steps are implemented using forexample, data analysis subsystem 203 and statistical analysis subsystem204 as previously detailed.

In step 704, the minority carrier population decay curves obtained instep 702 are analyzed to perform comparisons between samples. In oneembodiment, in step 704, methods of semiparametric testing are used todetect spatial nonuniformities in the semiconductor sample ordifferences between semiconductor samples. In one embodiment, asdescribed in, for example, p. 251-266 of N. Balakrishnan, and C. R. Rao“Handbook of statistics: advances in survival analysis. Vol. 23” AccessOnline via Elsevier, 2004. the Cox proportional hazard analysis model isused. This assumes that the discrete time hazard probabilitiesλ_(e)[(n+1)Δt] for the samples are proportional to each other. In afurther embodiment, results can be tested for validity of theproportional hazard assumption. Examples of tests for validity aredescribed in D. Schoenfeld, “Partial residuals for the proportionalhazards regression model.” Biometrika vol. 69 no 0.1 pp. 239-241 (1982);and T. M. Thernau et at “Modeling Survival Data: Extending the CoxModel” New York: Springer-Verlag 2000.

In another embodiment, in step 704, various nonparametric comparisontechniques can be used to analyse the p(t) obtained in step 702. In oneembodiment, the Mantel-Cox or logrank test is used, as described inSection 7.3 and 7.7 of Klein et at “Survival Analysis: Techniques forCensored and Truncated Data” Springer, 1997. In another embodiment, theGehan-Breslow test is used as described in the references E. A. Gehan,“A generalized Wilcoxon test for comparing arbitrarily singly-censoredsamples.” Biometrika vol 52, no. 1-2 pp. 203-223 (1965); and N. Breslow“A generalized Kruskal-Wallis test for comparing K samples subject tounequal patterns of censorship.” Biometrika vol. 57 no. 3 pp. 579-594(1970). In another embodiment, the Tarone-Ware test is used as describedin the reference R. E. Tarone et at “On distribution-free tests forequality of survival distributions.” Biometrika vol 64 no. 1 pp. 156-160(1977). In yet another embodiment, the tests proposed in T. R. Fleminget at “Counting processes and survival analysis” Vol. 169. Wiley.com,2011 are used. In another embodiment, one or more such comparisons areperformed, depending on, for example, whether the survival curves to becompared cross with each other or the requirements of the analysis. Inone embodiment, these tests are performed by statistical analysissubsystem 203. In another embodiment, these tests are performed by acombination of statistical analysis subsystem 203 and data analysissubsystem 204.

In yet another embodiment, in step 704, a combination of the previouslydescribed Cox proportional hazards analysis approach and thenonparametric approaches described above are used. Firstly, a visualcheck is performed to see if the discrete time hazard probabilities forthe samples cross. If not, then the validity of the assumption ofproportional hazards is tested. If the assumption of proportionalhazards is valid, then the logrank test is used. If the discrete timehazard probabilities cross, a different test is used, such as the testoutlined in A. Renyi “On the Theory of Order Statistics” ActaMathematica Hungarica vol. 4 pp. 191-231, 1953.

In a further embodiment, in step 704, one or more combinations ofanalyses are performed. For example, once the non-parametric tests havebeen performed and differences between the populations have beenobserved, then the summary statistics for each sample or each point canbe calculated and compared. An example flowchart is shown in FIG. 8.Steps 801-803 are similar to steps 601-603 respectively, and performedfor each sample or each point within a sample using, for example, dataanalysis subsystem 203 and statistical analysis subsystem 204 aspreviously detailed.

The advantage of the new analysis approaches over the previousparametric analysis approaches is that there are no assumptions of theform of the minority carrier population decay curve p(t). This thereforeovercomes the problems due to relying on the use of models withunrealistic assumptions. By using minority carrier population decaycurves p(t) or converting to a normalized decay curve and applying theunderstanding that this can be converted to a discrete time estimate ofthe CDF, methods of probabilistic analysis can be applied as describedabove without having to perform fitting to models which are inherentlyunrealistic.

In an additional embodiment, one or more “windows” of interest aredetermined. Each of these windows comprises a lower bound and an upperbound, and the decay curve between these bounds is extracted. Then oneor more statistical computations are performed using these one or morewindows. For example, in one embodiment the mean of the values withinthe window given by E[τ|n₁Δt≦τ≦n₂Δt]; n₁=0, 1, 2 . . . is computed usingknown formulas. Similarly other computations such as calculation ofvariance, Laplace transform, characteristic function, moments can alsobe performed. In another embodiment, the survival statisticalcomputations and the comparison of survival behavior techniques outlinedabove and in FIG. 7 are performed using these one or more windows.

Various methods can be used to determine the window size. In oneembodiment, in order to analyse the decay of the minority carrierpopulation within a localized region of interest surrounding the pointwhere minority carriers are generated by the incidence of light, awindow with lower bound t=0 and upper bound t=t_(win) is set. The upperbound can be set in a variety of ways.

In one embodiment, t_(win) is estimated by calculating the proportion ofrecombination events taking place within the localized region ofinterest compared to the number of recombination events taking placewithin the entire semiconductor sample, using one of the previouslyderived models such as in equations (1) and (2). An example is presentedin FIG. 9.

-   -   Step 901: Determining the region of interest R    -   Step 902: Using the model, determining the partial time        derivative of the minority carrier concentration

$\frac{\partial}{\partial t}\left\lbrack {p\left( {x,y,z,t} \right)} \right\rbrack$

-   -   Step 903: Spatially integrating

$\frac{\partial}{\partial t}\left\lbrack \left( {p\left( {x,y,z,t} \right)} \right\rbrack \right.$

within the region of interest R using, for example, a triple integral

$\int{\int{\int_{R}{\frac{\partial}{\partial t}\left\lbrack {p\left( {x,y,z,t} \right)} \right\rbrack}}}$

-   -   Step 904: Determining the proportion κ that the ∫∫∫_(R)        ∂t/∂[p(x, y, z, t)] comprises of the overall

${\frac{}{t}\left\lbrack {p(t)} \right\rbrack},$

that is

$\kappa = \frac{\int{\int{\int_{R}{\frac{\partial}{\partial t}\left\lbrack {p\left( {x,y,z,t} \right)} \right\rbrack}}}}{\frac{}{t}\left\lbrack {p(t)} \right\rbrack}$

-   -   Step 905: Determining a threshold proportion κ_(T)    -   Step 906: Denoting the time when κ drops below κ_(τ) as t_(win).

In another embodiment, t_(win) is estimated by using one of thepreviously derived models to estimate the proportion of minoritycarriers within the localized area as compared to the entiresemiconductor sample. An example is presented in FIG. 10:

-   -   Step 1001: Determining the region of interest R    -   Step 1002: Spatially integrating p(x, y, z, t) within region of        interest R using the triple integral ∫∫∫_(R) p (x, y, z, t)    -   Step 1003: Determining the proportion κ that the ∫∫∫_(R) p (x,        y, z, t) comprises of the overall p(t), that is

$\kappa = \frac{\int{\int{\int_{R}{p\left( {x,y,z,t} \right)}}}}{p(t)}$

-   -   Step 1004: Determining a threshold κ_(T) for the proportion    -   Step 1005: Denoting the time when κ drops below κ_(T) as        t_(win).

An example of the approach in FIG. 10 is provided in Chapter 7 of RameshRajaduray, “Investigation of Spatial Characterisation Techniques inSemiconductors,” Honours Thesis 1998, University of Western Australia,for the model described in W. Van Roosbroeck, “Injected Current CarrierTransport in a Semi-Infinite Semiconductor and the Determination ofLifetimes and Surface Recombination Velocities.” Journal of AppliedPhysics 26.4 (1955): 380-391 as explained earlier.

In another embodiment, t_(win) is determined using, for example,numerical simulations such as Monte Carlo simulations.

In another embodiment, t_(win) is determined using, for example,historical results from previous experiments or other types ofcharacterization techniques.

In an embodiment, the setting of t_(win) is performed using dataanalysis subsystem 204. In another embodiment, the setting of t_(win) isperformed using a combination of data analysis subsystem 204 andstatistical analysis subsystem 203.

In another embodiment, once t_(win) is known, then referring to FIG. 3the number of measurements (M) needed to perform a valid analysis isdetermined. Referring to FIG. 3, for example if a minimum of M_(min)samples are needed, then Δt (302) is set such that

${\Delta \; t} \leq \frac{t_{win}}{M_{\min}}$

In a further embodiment, clustering is performed. For example, in thecase where tests are performed to spatially characterize a semiconductorsample, different data points are grouped into spatial clusters based ondifferent clustering metrics. In one embodiment, the clustering metricis the probability that two samples are drawn from the same population,based on their survival curves. For example, referring to FIG. 4, if theprobability that the survival curves belonging to points 401 and 402 aredrawn from the same population is above a threshold, then it is likelythat points 401 and 402 have very similar parameters, that is, there isno spatial variation between these points. Then 401 and 402 belong tothe same cluster. However if the probability that the survival curvesbelonging to points 401 and 403 are drawn from the same population isbelow a threshold, then it is likely that there is spatial variationbetween points 401 and 403. Then points 401 and 403 do not belong in thesame cluster.

Continuing the above example, assume that 402 and 403 also belong in thesame cluster. Then two clusters for the points A, B and C can becreated:

-   -   Cluster 1: (401, 402)    -   Cluster 2: (402, 403)

This is an example of overlapping clusters, that is, where a pointbelongs to a plurality of clusters. In the example above, point 402belongs to clusters 1 and 2. It is also possible to stipulate thatclusters are non-overlapping, that is, where a point belongs to only onecluster. Then a given point will be assigned to the cluster which is theclosest match.

In one embodiment, the clustering is performed using pairwisecomparison, as demonstrated above. In another embodiment, the clusteringis performed on the basis of summary statistics. In another embodiment,clusters are pre-defined using the results of other tests.

In a further embodiment, the clustering demonstrated above is extendedto a plurality of semiconductor samples.

The methods explained above are not just limited to the transientphotoconductive decay technique. The methods can be extended to otherfields where a population is introduced and outputs related to the decaycurves of the introduced population are readily available formeasurement, so as to enable conversion into survival functions.

Although the algorithms described above including those with referenceto the foregoing flow charts have been described separately, it shouldbe understood that any two or more of the algorithms disclosed hereincan be combined in any combination. Any of the methods, algorithms,implementations, or procedures described herein can includemachine-readable instructions for execution by: (a) a processor, (b) acontroller, and/or (c) any other suitable processing device. Anyalgorithm, software, or method disclosed herein can be embodied insoftware stored on a non-transitory tangible medium such as, forexample, a flash memory, a CD-ROM, a floppy disk, a hard drive, adigital versatile disk (DVD), or other memory devices, but persons ofordinary skill in the art will readily appreciate that the entirealgorithm and/or parts thereof could alternatively be executed by adevice other than a controller and/or embodied in firmware or dedicatedhardware in a well known manner (e.g., it may be implemented by anapplication specific integrated circuit (ASIC), a programmable logicdevice (PLD), a field programmable logic device (FPLD), discrete logic,etc.). Also, some or all of the machine-readable instructionsrepresented in any flowchart depicted herein can be implemented manuallyas opposed to automatically by a controller, processor, or similarcomputing device or machine. Further, although specific algorithms aredescribed with reference to flowcharts depicted herein, persons ofordinary skill in the art will readily appreciate that many othermethods of implementing the example machine readable instructions mayalternatively be used. For example, the order of execution of the blocksmay be changed, and/or some of the blocks described may be changed,eliminated, or combined.

It should be noted that the algorithms illustrated and discussed hereinas having various modules which perform particular functions andinteract with one another. It should be understood that these modulesare merely segregated based on their function for the sake ofdescription and represent computer hardware and/or executable softwarecode which is stored on a computer-readable medium for execution onappropriate computing hardware. The various functions of the differentmodules and units can be combined or segregated as hardware and/orsoftware stored on a non-transitory computer-readable medium as above asmodules in any manner, and can be used separately or in combination.

While particular implementations and applications of the presentdisclosure have been illustrated and described, it is to be understoodthat the present disclosure is not limited to the precise constructionand compositions disclosed herein and that various modifications,changes, and variations can be apparent from the foregoing descriptionswithout departing from the spirit and scope of an invention as definedin the appended claims.

What is claimed is:
 1. A method for characterizing a semiconductorsample, said method implemented using an analysis system comprising: atransient photoconductive decay measurement subsystem, a database, adata analysis subsystem, and a statistical analysis subsystem, saidtransient photoconductive decay measurement subsystem, database, dataanalysis subsystem and statistical analysis subsystem connected to eachother by an interconnection, said method comprising the steps of:shining, using the transient photoconductive decay measurementsubsystem, light on one or more points in said semiconductor sample;measuring, using the transient photoconductive decay measurementsubsystem, one or more voltage decay curves corresponding to saidshining of light on said one or more points in said semiconductorsample; extracting, using the data analysis subsystem, one or moreintermediate voltage decay curves corresponding to one or more measuredvoltage decay curves; obtaining, using the data analysis subsystem, oneor more normalized decay curves corresponding to one or moreintermediate voltage decay curves, each of the said one or morenormalized decay curves corresponding to one or more discrete estimatesof survival functions; analyzing, using the statistical analysissubsystem, said obtained one or more normalized decay curves, saidanalyzing comprising obtaining one or more discrete estimates of theprobability of recombination corresponding to the one or more normalizeddecay curves, and computing one or more summary statistics correspondingto each of said obtained one or more discrete estimates; anddetermining, by either the statistical analysis subsystem or the dataanalysis subsystem, the presence of nonuniformities within saidsemiconductor sample based on results of said analyzing.
 2. A method forcharacterizing a semiconductor sample, said method comprising: measuringa plurality of voltage decay curves corresponding to a plurality ofpoints in said semiconductor sample; extracting a plurality ofintermediate voltage decay curves corresponding to said plurality ofmeasured voltage decay curves; converting said extracted plurality ofintermediate voltage decay curves to a plurality of minority carrierpopulation decay curves; performing one or more comparisons of survivalbehavior using said plurality of minority carrier population decaycurves; and determining presence of nonuniformities within saidsemiconductor sample using said results from said performing of one ormore comparisons of survival behavior.
 3. A method for comparing aplurality of semiconductor samples, said method comprising: measuring aplurality of voltage decay curves corresponding to a plurality of pointsfrom said plurality of semiconductor samples; extracting a plurality ofintermediate voltage decay curves corresponding to said plurality ofmeasured voltage decay curves; converting said extracted plurality ofintermediate voltage decay curves to a plurality of minority carrierpopulation decay curves; performing one or more comparisons of survivalbehavior using said plurality of minority carrier population decaycurves; and determining presence of nonuniformities within saidplurality of semiconductor samples using said results from saidperforming of one or more comparisons of survival behavior.